Imperfect production inventory model with uncertain elapsed time

  • Prasanta Kumar Ghosh Yogoda Satsanga Palpara Mahavidyalaya, Purba Medinipur, West Bengal, India
  • Jayanta Kumar Dey Mahishadal Raj College,Mahishadal, PurbaMedinipur, West Bengal, India
Keywords: Inventory, Imperfect production, Uncertain variables, Uncertain distribution, Expected value model


Most of the classical inventory control model assumes that all items received conform to quality characteristics. However, in practice, items may be damaged due to production conditions, transportation and environmental conditions. Modelling such real world problems involve various indeterminate phenomena which can be estimated through human beliefs. The uncertainty theory proposed by Liu (2015) is extensively regarded as an appropriate tool to deal with such uncertainty. This paper investigates the optimum production run time and optimum cost in an imperfect production process, where the rate of imperfect items are different in different states of the process. The process may be shifting from ‘in-control’ state to the ‘out-of-control’ state is an uncertain variable with certain uncertainty distribution. Some propositions are derived for the optimal production run time and optimized the expected total cost function per unit time. Finally, numerical examples have been illustrated to study the practical feasibility of the model.


Download data is not yet available.


Chen, C. K., & Lo, C. C. (2006). Optimal production run length for products sold with warranty in an imperfect production system with allowable shortages. Mathematical and Computer Modelling, 44, 319-331. DOI:

Chen, S. H., Wang, S. T., & Chang S. M. (2005). Optimization of fuzzy production inventory model with repairable defective products under crisp and fuzzy production quantity. International Journal of Operations Research, 2(2), 31-37.

Chen, L., Peng, J., Liu, Z., & Zaho, R. (2016). Pricing and effort decisions for a supply chain with uncertain information. International Journal of Production Research, 55(1), 264-284. DOI:

Chiu, S. W., Wang, S. L., & Chiu, Y. S. P. (2007). Determining the optimal run time for EPQ model with scrap, rework, stochastic breakdowns. European Journal of Operational Research, 180, 664-676. DOI:

Chiu, Y. S. P., Chen, K. K., Cheng, F. T., & Wu, M. F. (2010). Optimization of the finite production rate with scrap, rework and stochastic machine breakdown. Computers and Mathematics with Applications, 59, 919-932. DOI:

Gao Y., & Kar S., (2017). Uncertain solid transportation problem with product blending. International Journal of Fuzzy Systems, 19(6), 1916-1926. DOI:

Hiriga, M., & Ben Daya, M. (1998). The economic manufacturing lot-sizing problem with imperfect production process: Bounds and optimal solutions. Naval Research Logistics, 45, 423-432. DOI:<423::AID-NAV8>3.0.CO;2-7

Hu, F., & Zong, Q. (2009). Optimal production run time for a deteriorating production system under an extended inspection policy. European Journal of Operational Research, 196, 979-986. DOI:

Hussain, A. Z. M. O., & Murthy, D. N. P. (2003). Warranty and Optimal Reliability Improvement through Product Development. Mathematical and Computer Modelling, 38, 1211-1217. DOI:

Jiang, Y., Yan, H., & Zhu, Y. (2016). Optimal Control Problem for uncertain linear systems with multiple input delays. Journal of Uncertainty Analysis and Applications, 4, 1-15. DOI:

Kar, M. B., Majumder, S., Kar, S., & Pal, T. (2017). Cross-entropy based multi-objective uncertain portfolio selection problem. Journal of Intelligent & Fuzzy Systems, 32(6), 4467-4483. DOI:

Ke, H., Li, Y., & Huang ,H. (2015). Uncertain pricing decision problem in closed-loop supply chain with risk-averse retailer. Journal of Uncertainty Analysis and Application, 3, 1-14. DOI:

Ke, H., & Liu, B. (2010). Fuzzy Project Scheduling problems and its hybrid intelligent Algorithm. Applied Mathematical Modeling, 34(2), 301-308. DOI:

Khouja, M., & Meherez, A (1994). An economic production lot size model with imperfect quality and variable production rate. Journal of the Operational Research Society, 45, 1405-1417. DOI:

Liu, B. (2012). Why is there a need for uncertainty theory?. Journal of Uncertainty Systems, 6(1), 3-10.

Liu, B. (2015). Uncertainty theory: A branch of Mathematics for Modeling Human uncertainty. (4th ed.). Berlin: Springer-Verlag.

Liu, B. (2016). Uncertainty theory: A branch of Mathematics for Modeling Human uncertainty. In Liu, B Uncertainty Theory (pp. 1-79). Berlin: Springer-Verlag. DOI:

Liu, B. (2009). Theory and Practice of Uncertain Programming. (2nd ed.). Berlin: Springer-Verlag. DOI:

Liu, Y., & Ha, M. (2010). Expected value of function of uncertain variables. Journal of Uncertain Systems, 4(3), 181-186.

Liu,Y., Li, X., & Xiong, C. (2015). Reliability analysis of unrepairable systems with uncertain lifetimes. International Journal of Security and its Application, 9(12), 289-298. DOI:

Majumder, S., Kar, S., & Pal, T. (2018). Mean-Entropy Model of Uncertain Portfolio Selection Problem. In: JK Mandal, S, Mukhopadhyay, P, Dutta (Eds.), Multi-Objective Optimization: Evolutionary to Hybrid Framework. (pp. 25-54). Singapore: Spinger.

Majumder, S., Kundu, P., Kar, S., Pal, T. (2018). Uncertain multi-objective multi-item fixed charge solid transportation problem with budget constraint. In Majumder, S., Kundu, P., Kar, S., Pal, T (Eds.), Soft Computing (pp.3279-3301). Berlin: Springer. DOI:

Qin, Z., & Kar, S. (2013). Single-period inventory problem under uncertain environment. Applied Mathematics and Computation, 219, 9630-9638. DOI:

Qin Z., Kar, S., & Zheng, H. (2016). Uncertain portfolio adjusting model using semiabsolute deviation. Soft Computing, 20(2), 717-725. DOI:

Rosenbaltt, M. J., & Lee, H. L. (1986). Economic production cycles with imperfect production process. IIE Transactions, 17, 48-54. DOI:

Sana, S. S. (2010). An economic production lot size model in an imperfect production system. European Journal of Operational Research, 201, 158-170. DOI:

Sana, S. S. (2010). A production-inventory model in an imperfect production process. European Journal of Operational Research, 200, 451-64. DOI:

Wang, D., Qin Z., & Kar, S. (2015). A novel single period inventory problem with uncertain random demand and its application. Applied Mathematics and Computation, 269, 133-145. DOI:

Wang, X., & Tang, W. (2009). Optimal production run length in deteriorating production process with fuzzy elapsed time. Computer and Industrial Engineering, 56, 1627-1632. DOI:

Widyadana, G. A., & Wee, H. M. (2011). Optimal deteriorating items production inventory models with random machine breakdown and stochastic repair time. Applied Mathematical Modelling, 35(7), 3495-3508. DOI:

Yeh, R. H., Ho, W. T., & Tseng, S. T. (2000). Optimal production run length for products sold with warranty. European Journal of Operational Research, 120, 575-582. DOI:

Yeh, R. H., Chen, M. Y., & Lin, C. Y. (2007). Optimal periodic replacement policy for repairable products under free-repair warranty. European Journal of Operational Research, 176, 1678-1686. DOI:

Yeo, W. M., & Yuan, XueMing. (2009). Optimal warranty policies for systems with imperfect repair. European Journal of Operational Research, 199, 187-197. DOI:

Yun, Won Young., Murthy, D. N. P. & Jack, N. (2008). Warranty servicing with imperfect repair. International Journal of Production Economics, 111, 159-169. DOI:

Zhou, C., Tang, W., & Zhao, R. (2014). An uncertain search model for recruitment problem with enterprise performance. Journal of Intelligent Manufacturing, 28(3), 695-704. DOI:

How to Cite
Ghosh, P. K., & Dey, J. K. (2019). Imperfect production inventory model with uncertain elapsed time. Decision Making: Applications in Management and Engineering, 3(2), 1-18.